Maclaurin series by substitution
Problem 7.274 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = \frac{\sin{\left(2 x \right)}}{x} \).
- Start from a known series (sin u / x) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{\frac{8 x^{6}}{315} - \frac{4 x^{4}}{15} + \frac{4 x^{2}}{3} - 2 + \frac{\sin{\left(2 x \right)}}{x}}{x^{6}}\right) = 0 \]These terms match f through x^6.✓ Proved
Answer \( - \frac{8 x^{6}}{315} + \frac{4 x^{4}}{15} - \frac{4 x^{2}}{3} + 2 + \cdots \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | coefficients from derivatives at 0, and f − T is tiny at x = 0.01 |
Reviewers
gpt-oss:20b: fail (misleading) — The solution claims the series terms are 8x^6/315 − 4x^4/15 + 4x^2/3 − 2, which is the negative of the correct Maclaurin expansion. The reasoning is incomplete and could mislead a student into using the wrong signs.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the known Maclaurin series for sin(u) or show the substitution u=2x and division by x, which are the core steps required by the problem statement. It merely asserts a method and verifies the result algebraically, omitting the derivation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly state the known Maclaurin series for sin(u) or show the substitution u=2x and division by x, which are the core steps required by the problem statement. It merely asserts a method and verifies the result algebraically, omitting the derivation.gpt-oss:20b: fail (misleading) 2026-10-04 — The solution claims the series terms are 8x^6/315 − 4x^4/15 + 4x^2/3 − 2, which is the negative of the correct Maclaurin expansion. The reasoning is incomplete and could mislead a student into using the wrong signs.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer lists the terms in descending order of power (x^6, x^4, x^2, constant), which is not the standard form for a Maclaurin series (ascending order). More critically, the signs in the stated answer are incorrect: the expansion of sin(2x)/x is 2 - (4/3)x^2 + (4/15)x^4 - (8/315)x^6 + ..., but the answer gives - (8/315)x^6 + (4/15)x^4 - (4/3)x^2 + 2. While the set of terms is correct, the presentation as a sum implies these are the additive terms. The equation check confirms the terms match, but the 'Stated answer' format is ambiguous and potentially misleading regarding the sign of the x^6 term if read as a polynomial P(x) = ... - 8/315 x^6. Actually, looking closely: 2 - 4/3 x^2 + 4/15 x^4 - 8/315 x^6. The stated answer is: - 8/315 x^6 + 4/15 x^4 - 4/3 x^2 + 2. This is mathematically equivalent. However, the solution text is extremely sparse. The main issue is that the solution does not actually *derive* the terms, it just asserts a limit check. A student needs to see the substitution u=2x into sin(u) = u - u^3/3! + u^5/5! - ... and then division by x. The provided solution skips all derivation steps, making it useless for learning. Furthermore, the 'Stated answer' is not in ascending order of powers, which is non-standard for series expansions.gpt-oss:20b: pass 2026-10-04
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/maclaurin_substitution, checked 2026-10-04 with SymPy 1.14.0.